As the document indicates, |x| with a subscript of 2 corresponds to the L2 normal, which is the Euclidean Norm, also known as Euclidean distance. The Euclidean distance is like sqrt( (x1-x2)^2 + (y1-y2)^2) and so on. Notice that in the formula given, they square this value -- that is what the superscript 2 is about, squaring the result of the Euclidean distance. The squaring cancels out the sqrt(), leaving (x1-x2)^2 + (y1-y2)^2

The |x| with a subscript 1 is the L1 norm, also known as the taxicab distance or the Manhattan norm. This corresponds to x1-x2 + y1-y2 with no sqrt(). Notice though that in the formula they do not square this, leaving x1-x2 + y1-y2, abs(x1-x2) + abs(y1-y2)